Question

Let z and $\omega$be two complex numbers. If Re(z) = $|z-2|,Re\left(\omega \right)=|\omega -2|$ and arg (z $-$ $\omega$) = $\frac{\pi }{3}$, then Im (z + $\omega$)=

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

Re z = $|$z - 2$|$ $⟹x^2={(x-2)}^2+y^2$
$y^2=4(x-1)$
Parametric from z = 1 + $t^2+2ti$
Likewise, $\omega =1+s^2+2si$
arg (z $-$ $\omega$) = agr $(t^2-s^2+2(t-s)i)$
= ${tan}^{-1}\frac{2}{t+s}=\sqrt{3}\therefore t+s=\frac{2}{\sqrt{3}}$
$Im(z+\omega )$ = 2 (t + s) = $\frac{4}{\sqrt{3}}$